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Search results for: NONDIOPHANTINE ARITHMETIC AND CALCULUS

Fourier transforms on Cantor sets: A study in nonDiophantine arithmetic and calculus
PublicationFractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required basic properties. As an example we discuss a sawtooth signal on the ternary middlethird Cantor set. The formalism works also for fractals that are not selfsimilar.

Simple Fractal Calculus from Fractal Arithmetic
PublicationNonNewtonian calculus that starts with elementary nonDiophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a realvalued...

If Gravity is Geometry, is Dark Energy just Arithmetic?
PublicationArithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are nonunique. The examples of fourdimensional spaces, R^4 and (−L/2,L/2)^4, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a nonDiophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms...

NonNewtonian Mathematics Instead of NonNewtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics
PublicationNewtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh(tanh−1(β1)+tanh−1(β2)), although multiplication β1⊙β2=tanh(tanh−1(β1)⋅tanh−1(β2)),...

NonDiophantine Arithmetics in Mathematics, Physics and Psychology
PublicationFor a long time, all thought there was only one geometry — Euclidean geometry. Nevertheless, in the 19th century, many nonEuclidean geometries were discovered. It took almost two millennia to do this. This was the major mathematical discovery and advancement of the 19th century, which changed understanding of mathematics and the work of mathematicians providing innovative insights and tools for mathematical research and applications...

Marek Czachor prof. dr hab.
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BellType Inequalities from the Perspective of NonNewtonian Calculus
PublicationA class of quantum probabilities is reformulated in terms of nonNewtonian calculus and projective arithmetic. The model generalizes spin1/2 singlet state probabilities discussed in Czachor (Acta Physica Polonica:139 70–83, 2021) to arbitrary spins s. For s → ∞ the formalism reduces to ordinary arithmetic and calculus. Accordingly, the limit “nonNewtonian to Newtonian” becomes analogous to the classical limit of a quantum theory

Information Processing and Fechner’s Problem as a Choice of Arithmetic
PublicationFechner’s law and its modern generalizations can be regarded as manifestations of alternative forms of arithmetic, coexisting at stimulus and sensation levels. The world of sensations may be thus described by a generalization of the standard mathematical calculus.

Arithmetic Loophole in Bell's Theorem: Overlooked Threat to EntangledState Quantum Cryptography
PublicationBell’s theorem is supposed to exclude all local hiddenvariable models of quantum correlations. However,an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical oftwoelectron singlet states. Observable probabilities are consistent with the usual arithmetic employedby macroscopic observers...

Crystallization of space: Spacetime fractals from fractal arithmetic
PublicationFractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in spacetime can be generated by means of homogeneous spaces associated...

Application of nonclassical operational calculus to indicate hazards in numerical solutions of engineering problems
PublicationThe article addresses the application of non classical operational calculus to approximative solutions of engineering problems. The engineeringsound examples show that a continuous–discrete problem transformation from differential unequivocal problem to a differential wildcard problem, triggering a change in solution quality. A number of approximative methods are capable to alter both quantitative and qualitative...

Nonclassical operational calculus applied to certain linear discrete timesystem
PublicationW pracy zastosowano nieklasyczny rachunek operatorów do wyznaczania odpowiedzi pewnego dyskretnego układu dynamicznego. Pokazano metodę szczególnie przydatną do wyznaczania odpowiedzi pewnych dyskretnych niestacjonarnych układów dynamicznych, przy których zastosowanie przekształcenia Z sprawia duże trudności.

Unifying Aspects of Generalized Calculus
PublicationNonNewtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated.

Determination of the shape of the CFGFT cylindrical column based on laboratory tests
PublicationAnalyses were carried out on glassfibrereinforced polymer tube columns with reference to laboratory tests. The angles of the glass fibre beams were 20◦, 55◦ and 85◦. The study employed nonclassical operational calculus. Various modulated harmonic signal shapes were considered for columns and tubes at buckling. The buckling loads were assessed and compared for different models.

Impact of diffusion coefficient averaging on solution accuracy of the 2D nonlinear diffusive wave equation for floodplain inundation
PublicationIn the study, the averaging technique of diffusion coefficients in the twodimensional nonlinear diffusive wave equation applied to the floodplain inundation is presented. As a method of solution, the splitting technique and the modified finite element method with linear shape functions are used. On the stage of spatial integration, it is often assumed that diffusion coefficient is constant over element and equal to its average...

Simulation of Wave Propagation in Media Described by FractionalOrder Models
PublicationIn this paper, algorithms for simulation of the wave propagation in electromagnetic media described by fractionalorder (FO) models (FOMs) are presented. Initially, fractional calculus and FO Maxwell's equations are introduced. The problem of the wave propagation is formulated for media described by FOMs. Then, algorithms for simulation of the nonmonochromatic wave propagation are presented which employ computations in the time...

Marek Zienkiewicz dr inż.
PeopleDoctor engineer Marek Hubert Zienkiewicz is a graduate of the Faculty of Geodesy, Spatial Engineering and Construction at the University of Warmia and Mazury in Olsztyn. During his engineering, master's and doctoral studies he developed his scientific interests under the supervision of representatives of the Olsztyn geodetic compensatory calculus school. In 2011, he obtained the title of Master of Science in Geodesy and Cartography,...

Długie pociągi
PublicationWskazano, że przy wprowadzaniu długich pociągów pojawiają się możliwości, ale i ograniczenia różnej natury oraz problemy do rozwiązania. Inne w przypadku długich pociągów i inne w przypadku ciężkich pociągów. Pokazano długie pociągi, jako szansę na polepszenie bezpieczeństwa na drogach, poprzez ograniczenie na nich ruchu samochodów ciężarowych. Łatwiej będzie w praktyce realizować pomysł „Tiry na tory” Długie pociągi są szansą...

Numerical and quantitative analysis of HIV/AIDS model with modified AtanganaBaleanu in Caputo sense derivative
PublicationFractional calculus plays an important role in the development of control strategies, the study of the dynamical transmission of diseases, and some other reallife problems nowadays. The timefractional HIV/AIDS model is examined using a novel method in this paper. Based on the Atanganaconcept Baleanu’s of a derivative in the Caputo sense, the current modified fractional derivative operator uses singular and nonlocal kernels....

Transport of dangerous goods by rail, and threats to the subsoil of the railway surface in the event of a disaster
PublicationIn Poland, in 2020, the mass of dangerous goods (loads) transported by rail was 26 151.06 thousand tone. This translated into the performance of 8 899 691.89 thousand tone  km of transport performance. In 2020, these figures accounted for 11.72% of the weight of goods transported by rail. The situation is similar in other countries around the world. With such a large volume of transport of dangerous...